The role of mathematical modelling in guiding the science and economics of malaria elimination

Abstract

Unprecedented efforts are now underway to eliminate malaria from many regions. Despite the enormous financial resources committed, if malaria elimination is perceived as failing it is likely that this funding will not be sustained. It is imperative that methods are developed to use the limited data available to design site-specific, cost-effective elimination programmes. Mathematical modelling is a way of including mechanistic understanding to use available data to make predictions. Different strategies can be evaluated much more rapidly than is possible through trial and error in the field. Mathematical modelling has great potential as a tool to guide and inform current elimination efforts. Economic modelling weighs costs against characterised effects or predicted benefits in order to determine the most cost-efficient strategy but has traditionally used static models of disease not suitable for elimination. Dynamic mathematical modelling and economic modelling techniques need to be combined to contribute most effectively to ongoing policy discussions. We review the role of modelling in previous malaria control efforts as well as the unique nature of elimination and the consequent need for its explicit modelling, and emphasise the importance of good disease surveillance. The difficulties and complexities of economic evaluation of malaria control, particularly the end stages of elimination, are discussed.

Figure 1

A simple deterministic mathematical model of malaria transmission. The diagram shows the compartmental structure of the model with time-dependent variables: S, uninfected and non-immune; I_S, infected with no prior immunity; R, uninfected with immunity; I_R, infected with prior immunity; d_treat, average duration of treated infection; d_in, average duration of untreated infection. The model represents a situation where disease is being controlled using treatment of symptomatic/clinical malaria. Uninfected individuals become infected at a rate proportional to the overall prevalence of malaria infection. Recovery takes place either as treatment of clinical malaria at a given coverage or as natural clearance of the parasites. Immunity is assumed to be lost if immune uninfected individuals are not challenged for a given time period. The model equations and a detailed description are found elsewhere. The model was used to demonstrate the potential for combinations of interventions for elimination programmes. This model does not include explicit vector population and transmission dynamics, multiple levels of immunity, or the liver or asexual stages of infection. It is a simple representation of an extremely complex biological system and is for the purposes of understanding the more general behaviour of malaria transmission specifically during elimination and could be used as a first step by policy-makers for strategy planning for a few years ahead.

Figure 2

Deterministic (red line) versus stochastic (blue line) modelling of malaria elimination using a model previously published elsewhere. Only one run of the stochastic model is shown for clarity. The phases of the WHO malaria control-to-elimination continuum1 are indicated by the shaded background. The ‘limit of detection by surveillance’ indicates the number of cases below which a malaria surveillance programme is unlikely to detect any malaria, thus suggesting ‘apparent elimination’ (yellow circle). An arbitrary example is shown in the figure. Because of this detection limit, only the upper portion of the figure can be represented by surveillance data (‘data & model’), whereas the lower portion can only be represented by modelling predictions (‘model only’). Improving the sensitivity of surveillance would lower this detection limit. For ‘true elimination’ to occur (green circle), the number of malaria cases must fall below the ‘elimination threshold’ (<1 case). Only a perfect surveillance system detecting every case would have a limit of detection by surveillance equal to the elimination threshold. This is not generally the case in the field where surveillance systems are far from perfect and can miss many cases. Thus, the limit of detection by surveillance is generally above the elimination threshold. If malaria control interventions are stopped inappropriately early when apparent elimination occurs (red dotted line), numbers of cases begin to increase again. Modelling gives an indication of how long control measures would need to be continued to achieve true elimination. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)